LECTURE 22. Collective effects in multi-particle beams: Parasitic Losses. Longitudinal impedances in accelerators (continued)

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1 LECTURE Collective effect in multi-particle beam: Longitudinal impedance in accelerator Tranvere impedance in accelerator Paraitic Loe /7/0 USPAS Lecture Longitudinal impedance in accelerator (continued) The broad-band reonator model The vacuum chamber of a typical accelerator i not a perfectly mooth round pipe. Diagnotic device, uch a beam poition monitor, are typically prinkled throughout the machine; thee device may have pickup plate and thu deviate from a cylindrical geometry. Special magnet, uch a kicker and epta for injection and extraction, or wiggler and undulator, may have irregular aperture. Special device uch a eparator, and the tranition into and out of rf cavitie, alo repreent change in the dimenion of the vacuum chamber. A very crude model for thee dicontinuitie in the vacuum chamber dimenion i to conider them to be mall reonant cavitie, of the following generic form /7/0 USPAS Lecture b Such a cavity ha a radiu b and a reonant frequency of order ω c R = b. In travelling pat thi cavity, the beam wake field that penetrate the cavity are left behind a the beam exit the cavity: thi contitute an energy lo to the beam. Thi may be etimated by computing the tored energy in the cavity due to the beam field. A roughly equal amount of energy at ω>c/b propagate down the pipe with the beam. Equating the total /7/0 USPAS Lecture 3 b energy lot by the beam to the integrated power lo on the cavity impedance give a crude etimate of the cavity impedance cloe to reonance: about 60 Ω. Examination of the repone of the beam to the cavity at low frequencie then how that the effective Q i cloe to. Thi i the bai of the broad-band reonator model. In thi model, the generic cavity i treated a a ingle, low-q reonator (Q=), with a reonant frequency ω c R = b, where b i the radiu of the vacuum chamber, and a hunt impedance R =60 Ω. From the general form for a cavity reonator, the impedance i then 0 60 Ω ( ω) = c bω + i bω c A plot i given below, for the cae b=3 cm: /7/0 USPAS Lecture 4

2 Re»» HWL Im»» HWL W0 HzL HVêpCL f HGHzL z HcmL -0 The impedance i peaked at a high frequency, about GHz. It tend to be motly reitive there, and motly inductive at low frequencie. The wake function i It ha quite a hort range, becaue of the low Q. It amplitude at mall z i comparable to the wake function from a narrow band reonator. The broad band reonator model i not very accurate for frequencie above cutoff ω >> c b ( z << b ), o the detail near z=0 are wrong. Neverthele, the model i ueful for rough etimate. /7/0 USPAS Lecture 5 /7/0 USPAS Lecture 6 We ll ee later (perhap) that the effect of the longitudinal impedance 0 on the dynamic of the beam cale like 0 ( ω) n ( ω ), where n( ω) = ω ω, with ω 0 being the revolution 0 frequency: ω π π 0 = T = c C. A broad band reonator thu give a contribution to the total 0 n of the machine equal to 0 Rω 0 377b n = Ω. A well-deigned machine will bb ω R C have a broad band impedance of no more than about 0 n Ω. Thu, the maximum number of generic max broad-band cavitie allowed per unit length i about nbb cavity C 377 b. For example, for b=3 cm, we need to have le than about uch cavity every m. Thi give a crude etimate of the required moothne of the machine vacuum chamber. Impedance of the reitive wall A relativitic point charge travelling through a vacuum chamber with perfectly conducting wall leave behind no wake field, ince the field do not penetrate the chamber. No energy i diipated in the wall. However, if the vacuum chamber wall have a finite conductivity, then energy will be diipated by the beam induced current, and a wake field will be produced. /7/0 USPAS Lecture 7 /7/0 USPAS Lecture 8

3 The full expreion for the wake field and wake potential can only be obtained by olving Maxwell equation in the reitive pipe. (See text, ec. 6.3.) However, we can get a crude etimate of the impedance of the wall in the following imple picture: δ b Let the conductivity of the wall be σ. The current flowing in a ection of the wall of length L pae through an area L A= πbδ, where b i the pipe radiu, and the kin depth iδ =. Thu, the reitance per unit length i σµω R wall µω = = = L σa σπbδ πb σ The full olution for the field how that the impedance i complex; the above i it real part. The complete expreion, for a machine of circumference C, i i 0 gn( ω) µω ( ω) = C πb σ The following plot how the longitudinal reitive wall impedance, for b=3 cm, an aluminum wall, and C=750 m. /7/0 USPAS Lecture 9 /7/0 USPAS Lecture 0 Re»» HWL Im»» HWL f HGHzL The aociated wake field can be etablihed by an invere Fourier tranform and i given by the equation W z = C c cµ 0() 4πb πσ 3 z It i plotted in the next figure W0 HzL HVêpCL z HcmL The low decay of the reitive wall wake function with z lead to a long tail. Total longitudinal impedance: The next plot how the total impedance 0 n a a function of frequency for 4 narrow band cavitie at 500 MHz (with the parameter given in the previou numerical example) and /7/0 USPAS Lecture /7/0 USPAS Lecture

4 generic broad band reonator. The reitive wall impedance i alo included, although it i mall: a few tenth of an ohm fhghzl /7/0 USPAS Lecture 3 Re Im ÅÅÅÅÅÅÅÅÅ»» n ÅÅÅÅÅÅÅÅÅ»» n At high frequencie, the impedance i motly reitive, dominated by the broad band reonator. Near the frequency of the rf cavitie, they dominate. At low frequencie, the impedance i motly inductive, due to the broad band reonator. One type of longitudinal impedance that we have not dicued here i the longitudinal pace charge impedance. The wake function are derivable from the longitudinal pace charge force, which reult from variation in the longitudinal charge denity. Like tranvere pace charge force, the wake function and the impedance decreae with /γ, and o are inconequential for high energy electron machine, but may play an important role in relatively low energy (-0 GeV) proton machine. Longitudinal pace charge i dicued in the text, ec 6... /7/0 USPAS Lecture 4 Tranvere impedance in accelerator The principal ource of tranvere impedance in accelerator are imilar to the longitudinal one that we have jut dicued. There will be tranvere impedance aociated with narrow band rf cavitie, broad band reonator, and the reitive wall. Narrow-band tranvere impedance For any mode m, the tranvere and longitudinal rf cavity impedance are related by m ω ( ) c ω = m( ω) Thu c c R ( ω) = ( ω) = ω ω ω + iq R ω ω ω in which the parameter R, Q and ω R now refer to a tranvere cavity mode, that i, one for which the field produce tranvere force. A plot of the tranvere impedance i: Re Im T wr ÅÅÅÅÅÅÅÅÅÅÅÅ c RS T wr ÅÅÅÅÅÅÅÅÅÅÅÅ c RS w ÅÅÅÅÅÅ wr R -0.5 /7/0 USPAS Lecture 5 /7/0 USPAS Lecture 6

5 The wake function for thi impedance can be obtained by taking a Fourier tranform. The reult, for z<0, i cr z z W z R ω R ( ) = exp in ω 4 Q cq c ( ) 4 ( Q ) ( ) z v λ rf Plot of W () z 4 Q cr for Q =0. è!!!!!!!!!!!!!!! W 4 Q - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ cr A for the longitudinal cae, the wakefield ocillate in z with a wavelength equal to λ rf ; it i damped to /e in a ditance Q rf π λ. Tranvere broad-band reonator We can model the tranvere effect of a generic cavity in the machine with the ame broad-band reonator model we ued in z ÅÅÅÅÅÅÅÅ lrf /7/0 USPAS Lecture 7 /7/0 USPAS Lecture 8 the longitudinal plane. To relate the tranvere impedance of a broad-band reonator to the longitudinal impedance, we ue the approximate reult quoted in Lecture 4: c 0( ω) ( ω) ωb. The broad-band tranvere impedance i then ( ω) 60 Ω c b ω c b + i ω bω c A plot i given below, for the cae b=3 cm: - - Re THkWêmL Im THkWêmL f HGHzL At low frequencie, the imaginary part dominate. The wake function i WHzL HVêpCêmL - -4 z HcmL /7/0 USPAS Lecture 9 /7/0 USPAS Lecture 0

6 It ha quite a hort range, becaue of the low Q. Numerical example: If a bunch paing through thi broad band reonator ha x0 particle, and it i off-axi in the cavity by cm in x, then the tranvere deflecting (integrated) force it produce at z~-5 cm i F x QW 0 ( 5cm) x = ( Ne) e 9 ( ). 0 V 7 kv V C The change in the x-trajectory lope of a particle following in the wake i F x = x F = x pv mcβγ 0 For an electron in CESR, with γ=0 4, we find 7 kev x = = 4 µrad ev 3 4. Thi i the effective peak dipole kick applied to a trailing particle by the wakefield of the bunch. /7/0 USPAS Lecture /7/0 USPAS Lecture Tranvere impedance of the reitive wall The reitive wall tranvere impedance can be obtained from the reitive wall longitudinal impedance uing the approximate relation from Lecture 4: c 0( ω) ( ω) ωb Uing the expreion given above for the reitive wall longitudinal impedance, we have for the tranvere impedance, for a machine of circumference C, in which an extra factor of ha been inerted in the numerator (the approximate relation given above i only good to a factor of two in thi cae). The following plot how the tranvere reitive wall impedance, for b=3 cm, an aluminum wall, and C=750 m Re THkWêmL Im THkWêmL ign( ω) ( ω) = C πb 3 µ c ωσ f HGHzL /7/0 USPAS Lecture 3 /7/0 USPAS Lecture 4

7 Thi impedance i quite trong at low frequencie. cc cµ The aociated wake field i given byw ()= z π 3 b πσ It i plotted in the next figure WHzL HVêpCêmL z HcmL z very long tail can be important in driving tranvere intabilitie in which multiple bunche are coupled together. Total impedance: The next plot how the total impedance a a function of frequency for 50 generic broad band reonator, and the reitive wall. Narrow band cavitie are not included; generally they do not play an important role, unle they have very trong tranvere deflecting mode. The decay of the reitive wall tranvere wake function with z i even lower than that of the longitudinal reitive wall. The /7/0 USPAS Lecture 5 /7/0 USPAS Lecture Re T HkWêmL Im T HkWêmL fhghzl tranvere pace charge force, obtained in lecture 3. The wake function and the impedance decreae with /γ, and o are inconequential for high energy electron machine, but may play an important role in relatively low energy (-0 GeV) proton machine The real part i dominated by broad band reonator at high frequencie, and the reitive wall at low frequencie. The imaginary part i motly due to the broad band reonator except at very low frequencie, where the reitive wall take off. Tranvere pace charge can alo be conidered to be a ource of impedance. The wake function are derivable from the /7/0 USPAS Lecture 7 Paraitic Loe When a bunch pae through a cavity or other ource of longitudinal impedance in a machine and generate longitudinal wakefield, thee field will tend to decelerate the bunch itelf. Such energy loe are called paraitic loe. Conider an extended charge ditribution ρ( ) paing through a cavity. An increment of charge dq = ρ( ) d in the front of the bunch /7/0 USPAS Lecture 8

8 produce a longitudinal wake function W0( z), which i een by an element of charge dq = ρ( ) d later in the bunch. ρ( ) d ρ( d ) z The incremental wake potential een by dq due to dq i d F = dq dq W ( z) 0 The total change in the energy of the bunch i E = d F = d ρ( ) dρ( ) W ( ) 0 Suppoe that the bunch i very, very hort: much horter than the ditance cale over which W0( z) varie. Then 0 0 E W ( ) d ρ( ) dρ( ) in which W 0 ( 0 ) i the value of the longitudinal wake function at a very mall ditance from z=0. Then, if we make the ubtitution u ( ) = ρ( d ) du= ρ( ) d u( ) = ρ( ) d = q /7/0 USPAS Lecture 9 /7/0 USPAS Lecture 30 where q i the total charge, then q d ρ( ) dρ( ) = udu = 0 q and E W 0 ( 0 ) We ee that for a very hort bunch (i.e., a point charge), the energy lot in paing through an impedance i one-half of the product of the charge quared with the longitudinal wake field produced by the point charge at z=0_. Thi i called the fundamental theorem of beam loading. For an rf cavity, from the expreion given above for the wake function, we have for q the paraitic energy lo of a point charge in the cavity q R E ω = R = qk Q R in which k ω = R i called the lo factor of the cavity. If the Q cavity can ocillate in mode other than the fundamental, there will be a k for each mode. Each k will give the energy depoited into that mode by a point charge travelling through the cavity, and will alo be related, by k = W 0 ( 0 ), to the wake function aociated with the impedance of that mode. Example: conider the 500 MHz narrow-band rf cavity dicued earlier. The lo factor i /7/0 USPAS Lecture 3 /7/0 USPAS Lecture 3

9 9 6 ω R k = R π = J V 4 0 = 04 Q C pc So a beam with a very hort bunch and a charge of = 3. 0 pc will looe about.8 kev in the cavity on each paage. In general, for a bunch of finite length, the paraitic energy lo will be le than for a point charge. The lo can be computed from the relation given above E = d ρ ( ) dρ ( ) W0 ( ) in which the lower integration limit ha been extended to, ince W 0 (z)=0 for z>0. Then, introducing the longitudinal impedance 0 0 ωz ( ω) = dzw ( z)exp i, thi expreion c c can be tranformed into ( ) E = dωρω ( ) Re 0( ω) π where ρω ( ) = dexp( iω) ρ( ) i the Fourier tranform of the longitudinal charge denity. For a Gauian bunch of charge Ne and rm length σ, we have ωσ ρω ( ) = exp Ne c /7/0 USPAS Lecture 33 /7/0 USPAS Lecture 34 o the paraitic lo i ( ) Ne E = ( ) d ωσ ω exp Re 0( ω) π c For a narrow band reonator in a ynchrotron, the wake field may lat more than one revolution. In thi cae, the wake field from previou bunch paage mut be included in the calculation of the paraitic energy lo. The expreion for the energy lo in thi cae become E = d ρ( ) dρ( ) W ( kc + ) k= 0 where C i the circumference. For a point charge q, thi i jut E = q W0( kc). k= It turn out that thi um can be done analytically for the cae of a reonator wake function, a given in Lecture 4, p. 33, for the cae of Q>>, and for the on-reonance cae ω R = hω 0 : πh E q cr exp + πh Q = C Q πh exp Q For πh Q <<, thi become jut E q cr = C /7/0 USPAS Lecture 35 /7/0 USPAS Lecture 36

10 Example: for the 500 MHz narrow-band rf cavity dicued earlier, if we evaluate the um over wake function on previou turn, we find, for C= hλ rf, with h=8, and with q = pc, cr E = q 00 kev C So the effect of the previou turn wake in the cavity in fact i much larger than the k=0 term, which wa etimated above at about.8 kev. The paraitic energy lo for a bunch of finite length, with longitudinal charge denity ρ(), including the effect of multiple turn, in term of the impedance, i = ( ) ω E 0 ρ pω0 ( ) Re 0( pω0) π p= For a point charge q, thi become = ( ) ω E q 0 Re 0( pω0) π p= and for a narrow band impedance, with p=h, Q>> and h/q<<, and hω 0 = ω R, we have E = q ω 0R π in agreement with the um over wake function given above. /7/0 USPAS Lecture 37 /7/0 USPAS Lecture 38